A comparison between stall prediction models for axial flow compressors

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Compressors

by

Andrew Gill

Thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mechanical Engineering at

Stellenbosch University

Department of Mechanical Engineering, University of Stellenbosch,

Private Bag X1, 7602 Matieland, South Africa.

Supervisors:

Prof. T.W. Von Backstr¨om Prof. G.D. Thiart

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Declaration

I, the undersigned, hereby declare that the work contained in this thesis is my own original work and that I have not previously in its entirety or in part submitted it at any university for a degree.

Signature: . . . . A. Gill

Date: . . . .

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Abstract

The Stellenbosch University Compressor Code (SUCC) has been developed for the purpose of predicting the performance of axial flow compressors by means of axisymmetric inviscid throughflow methods with boundary layer blockage and empirical blade row loss models. This thesis describes the process of the implementation and verification of a number of stall prediction criteria in the SUCC. In addition, it was considered desirable to determine how certain factors influence the accuracy of the stall prediction criteria, namely the nature of the computational grid, the choice of throughflow method used, and the use of a boundary layer blockage model and a radial mixing model. The stall prediction criteria implemented were the diffusion factor limit criterion, de Haller’s criterion, Aungier’s blade row criterion, Aungier’s boundary layer separation criterion, Dunham’s, Aungier’s and the static-to-static stability criteria. The compressors used as test cases were the Rofanco 3-stage low speed compressor, the NACA 10-stage subsonic compressor, and the NACA 5-stage and 8-stage transonic compressors. Accurate boundary layer blockage modelling was found to be of great importance in the prediction of the onset of stall, and that the matrix throughflow Method provided slightly better accuracy than the streamline curvature method as implemented in the SUCC by the author. The ideal computational grid was found to have many streamlines and a small number of quasi-orthogonals which do not occur inside blade rows. Radial mixing modelling improved the stability of both the matrix throughflow and streamline curvature methods without significantly affecting the accuracy of the stall prediction criteria. De Haller’s criterion was over-conservative in estimating the stall line for transonic conditions, but more useful in subsonic conditions. Aungier’s blade row criterion provided accurate results on all but the Rofanco compressor. The diffusion factor criterion provided over-optimistic predictions on all machines, but was less inaccurate than de Haller’s criterion on the NACA 5-stage transsonic machine near design conditions. The stability methods performed uniformly and equally badly, supporting the claims of other researchers that they are of limited usefulness with throughflow simulations. Aungier’s boundary layer separation method failed to predict stall entirely, although this could reflect a shortcoming of the boundary layer blockage model.

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Opsomming

Die Stellenbosch University Compressor Code (SUCC) is ontwikkel om die prestasie van ak-siaalvloei kompressors te voorspel met behulp van aksisimmetriese nie-viskeuse deurvloeime-todes met grenslaagblokkasie en empiriese modelle vir die verliese binne lemrye. Hierdie tesis beskryf die proses waarmee sekere staakvoorspellingsmetodes in die SUCC ge¨ımplementeer en geverifieer is. Dit was ook nodig om die effek van sekere faktore, naamlik die vorm van die berekeningsrooster, die keuse van deurvloeimetode en die gebruik van ‘n grenslaag-blokkasiemodel en radiale vloeivermengingsmodel op die akuraatheid van die staakvoor-spellingsmetodes te bepaal. Die staakvoorstaakvoor-spellingsmetodes wat ge¨ımplementeer is, is die diffusie faktor beperking metode, de Haller se metode, Aungier se lemrymetode, Aungier se grenslaagmetode en die Dunham, Aungier en die statiese-tot-statiese stabiliteitsmetodes. Die kompressors wat gebruik is om die metodes te toets is die Rofanco 3-stadium lae-spoed kompressor, die NACA 10-stadium subsoniese kompressor en die NACA 5- en 8-stadium transsoniese kompressors. Daar is vasgestel dat akkurate grenslaagblokkasie modelle van groot belang was om ‘n akkurate aanduiding van die begin van staking te voorspel, en dat, vir die SUCC, die Matriks Deurvloei Metode oor die algemeen ’n bietjie meer akkuraat as die Stroomlyn Kromming Metode is. Daar is ook vasgestel dat die beste berekeningsrooster een is wat baie stroomlyne, en die kleinste moontlike getal quasi-ortogonale het, wat nie binne lemrye geplaas mag word nie. Die numeriese stabiliteit van beide die Matriks Deurvloei en die Stroomlyn Kromming Metode verbeter deur gebruik te maak van radiale vloeiver-mengingsmodelle, sonder om die akkuraatheid van voorspellings te benadeel. De Haller se metode was oorkonserwatief waar dit gebruik is om die staak-lyn vir transsoniese vloei toestande, maar meer nuttig in die subsoniese vloei gebied. Aungier se lemrymetode het akkurate resultate gelewer vir alle kompressors getoets, behalwe die Rofanco. Die diffusie faktor metode was oor die algemeen minder akuraat as Aungier se metode, maar meer akku-raat as de Haller se metode vir transsoniese toestande. Die stabiliteitsmetodes het almal ewe swak gevaar. Dit stem ooreen met die bevindings van vorige navorsing, wat bewys het dat hierdie metodes nie toepaslik is vir simulasies wat deurvloeimetodes gebruik nie. Aungier se grenslaagmetode het ook baie swak gevaar. Waarskynlik is dit as gevolg van tekortkomings in die grenslaagblokkasiemodel.

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Acknowledgements

I would like to thank my two supervisors, Professor T.W. von Backstr¨om and Pro-fessor G.D. Thiart for their constant guidance, advice and support, and the innu-merable ways in which they have helped me to complete this project. I would also like to thank Thomas Roos for his advice and encouragement and his assistance in obtaining geometric data. I am also indebted to Adriaan Steenkamp, who also as-sisted me greatly in obtaining geometric data. I would also like to thank my family and friends for their emotional support and prayers. Finally, I am most grateful to Armscor, who provided me with financial support.

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List of Figures

2.1 Blade cascade nomenclature . . . 15 2.2 A simplified diagram of stall inception in a rotor blade row . . . 20 2.3 A fictitious total-to-static compressor characteristic with the 2-dimensional

stability stall point indicated . . . 25 3.1 Some blade row geometry factors used for Aungier’s blade row criterion 35 4.1 Computational grid for the Rofanco 3-stage compressor . . . 45 4.2 Predicted and experimental characteristics for the Rofanco 3-stage

com-pressor with and without boundary layers using MTFM . . . 47 4.3 Predicted and experimental characteristics for the Rofanco 3-stage

com-pressor using SCM with smooothing . . . 48 4.4 Computational grid for the NACA 5-stage compressor . . . 49 4.5 Predicted characteristics for the NACA 5-stage compressor with and

without boundary layer blockage effects . . . 53 4.6 Predicted efficiency characteristics for the NACA 5-stage compressor . 54 4.7 Predicted and experimental characteristics for the NACA 5-stage

com-pressor using MTFM . . . 55 4.8 Predicted and experimental characteristics for the NACA 5-stage

com-pressor using SCM with radial entropy smoothing . . . 56 4.9 Predicted and experimental characteristics for the NACA 5-stage

com-pressor using MTFM with 5 and 9 streamlines . . . 57 4.10 Predicted and experimental characteristics for the NACA 5-stage

com-pressor using SCM with and without radial entropy smoothing . . . . 58 4.11 Computational grid for the NACA 8-stage compressor . . . 59 4.12 Predicted characteristics for the NACA 8-stage compressor without

boundary layer blockage effects . . . 61 4.13 Computational grid for the NACA 10-stage compressor . . . 62

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4.14 Predicted characteristics for the NACA 10-stage compressor with and without boundary layer blockage effects . . . 64 4.15 Predicted efficiency characteristics for the NACA 10-stage compressor 65 4.16 Predicted and experimental characteristics for the NACA 10-stage

com-pressor using MTFM . . . 66 4.17 Predicted and experimental characteristics for the NACA 10-stage

com-pressor using SCM . . . 67 4.18 Predicted and experimental characteristics for the NACA 10-stage

com-pressor using MTFM with 7 and 9 streamlines . . . 68 4.19 Predicted and experimental characteristics for the NACA 10-stage

com-pressor using SCM with radial entropy smoothing . . . 69 B.1 The positioning of quasi-normals and streamlines on a compressor annulus 97

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List of Tables

4.1 Errors for stall prediction for the Rofanco compressor . . . 72 4.2 Stall-prediction errors for the NACA 5-stage compressor simulated with

5 streamlines and the MTFM . . . 72 4.3 Stall-prediction errors for the NACA 5-stage compressor with 9

stream-lines and the MTFM . . . 73 4.4 Stall-prediction errors for the NACA 5-stage compressor with 5

stream-lines and the SCM with radial smoothing . . . 73 4.5 Stall-prediction errors for the NACA 10-stage compressor with 7

stream-lines and the SCM with radial smoothing . . . 75

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Nomenclature

a blade profile dependent parameter for calculating Deq

as Speed of sound

c blade profile chord length

cp non-dimensional pressure coefficient

h static enthalpy

i blade cascade incidence angle

lc length of compressor in Moore and Greitzer (1986a)

m meridional coordinate ˙

m massflow rate

r radial coordinate

s entropy

t blade profile thickness

tb maximum blade profile thickness

w work input or output into a turbomachine

x axial coordinate in Moore and Greitzer (1986a)

y quasi-normal (quasi-orthogonal) coordinate

z axial coordinate

Ac frontal area of compressor in Moore and Greitzer (1986a)

B Mach-number dependent parameter in Moore and Greitzer (1986a)

C absolute velocity

DF diffusion factor

Deq equivalent diffusion factor

H stagnation enthalpy, boundary layer shape factor

I rothalpy M Mach-number P pressure T temperature U blade velocity V absolute velocity

Vp plenum chamber volume in Moore and Greitzer (1986a)

W relative velocity

WRE square-root of the relative outlet and inlet dynamic

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α angle of attack for blade cascade

β blade profile relative leading or trailing edge flow angle, radial mixing parameter

γ blade cascade stagger angle

² angle between quasi-normal and true normal

θ camber angle, wheel angle in Moore and Greitzer (1986a)

η non-dimensional axial coordinate in Moore and Greitzer (1986a)

ξ non-dimensional time unit in Moore and Greitzer (1986a)

ρ density

κ blade cascade metal angle

κm streamline curvature

λ quasi-normal angle

φ angle between meridional vector and axial direction, flow coefficient

ψ stream function, load coefficient

σ blade row solidity

ω angular velocity, blade row total pressure loss coefficient Subscripts

m meridional component of quantity

θ circumferential component of quantity rel relative property

0 total or stagnation thermodynamic conditions 1 blade row inlet property

2 blade row outlet property TT total-to-total

TS total-to-static SS static-to-static Superscripts

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Introduction

This chapter presents some of the background information necessary to understand the significance this project. A brief history of the axial compressor research con-ducted at the University of Stellenbosch is presented, so as to explain the origins and need for this project. The scope for this project is then defined, and the questions which this project has addressed are presented.

1.1 Background

This section will briefly explain the importance of stall prediction in the general field of axial flow compressor simulation, and in South Africa today. The origins of this project are also described.

1.1.1 General background

With the increasing speed of computer hardware and the increasing power of Com-putational Fluid Dynamics (CFD) software, it has become possible to simulate the flow phenomena within an axial flow compressor in three dimensions, taking viscous effects into account, as in Pratap et al. (2005), or in a time dependent effects, such as in Gr¨uber and Carstens (2001). However, the amount of labour and computational time required for such a project is still rather large, as was commented on by Pratap

et al. (2005). Furthermore, a large amount of experience and a high level of training

is required of the researcher operating the software if meaningful results are to be obtained. For these reasons the author believes that simpler simulation methods are often more suitable for many purposes. Finally, it is often considerably easier to make use of empirical relations and data in a custom written simulation than to try to incorporate them into a standard commercial CFD package.

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One family of methods that is frequently dealt with in literature such as Novak (1967), Cumpsty (1989), Gannon and von Backstr¨om (2000) and Aungier (2003), are the so-called throughflow methods. These typically make use of axisymmetric, inviscid flow approximations in which viscous effects such as endwall boundary layers are corrected for by blockage modelling. All blade-related phenomena and other loss mechanisms must also be simulated by means of separate models. Compressor stall is one such phenomenon.

Stall is a potentially dangerous mode of operation which should be prevented if possible as it can lead to damage of the compressor. This is because a compres-sor operating in a state of stall experiences significant fatigue loadings, which can damage or destroy the rotor and stator blades of the machine. Because of the as-sumptions made in the throughflow methods and the complex nature of flow in a stalled compressor, it is not apparent from the results whether a compressor being simulated would be undergoing stall or not. Thus separate stall criteria must be de-veloped, so as to ensure that the results are realistic. The stall criteria also provide a constraint or limit in the design of a compressor. However, the phenomenon of stall in axial flow compressors is still not fully understood.

Although there is some similarity between the phenomenon of stalling on an aircraft wing and stall in an axial flow compressor, the latter phenomenon is con-siderably more complicated. This is emphasised in most of the works read by the author containing a description of stall phenomena in compressors, especially Cump-sty (1989) and Pampreen (1993). Unlike an aircraft wing, the relative flow velocity seen by a rotor blade varies across its span in an axial flow compressor, because it is rotating relative to the flow and because of hub and shroud boundary layers. In addition, the entry conditions for a compressor stage are dictated by the nature of flow from the previous stage, and a stalled stage in a compressor will thus adversely affect the performance of stages following it, although not necessarily to the point where they too will begin to develop stall cells.

Many of the phenomena associated with the propagation of stall after its in-ception are thus highly three-dimensional in nature. This is made very clear in explanations of stall provided by researchers such as Day (1993). For this reason, they cannot be modelled effectively using axisymmetric methods. Furthermore, the main motivation behind the attempt to predict the onset of stall is so that it can be avoided. The stall prediction methods to be investigated here therefore aim to predict the range of operating conditions under which stall or surge will occur in a given compressor, while much of the research into the three-dimensional propagation of stall within a machine is conducted by means of practical testing of experimental compressor rigs.

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1.1.2 Background to this project

The South African Air Force requires the ability to predict the performance and life-expectancy of the various gas-turbine engines of the aircraft it operates. This is necessary for a number of reasons: the manufacturers of the aircraft engines are not always willing to provide the necessary information, performance data for the engine operating under South African conditions may not exist, or the engines may be very old, and nearing the end of their operational life. An important prerequisite to the development of this ability is that it should be possible to accurately predict the performance map for the axial flow compressors used in these engines. The development of this aspect of gas-turbine engine modelling and research has been entrusted to the University of Stellenbosch.

The University of Stellenbosch possesses a Rofanco 3-stage low-speed compressor test bench. At some point, the blades of this machine were destroyed during testing. Bernad´e (1986) describes the development of a computer software package which was used to design new blades. The new blades gave the machine a higher reaction ratio, and eliminated the need for inlet guide vanes.

The new blades were dully manufactured by the Atomic Energy Corporation. Lewis (1989) began the process of recommissioning the test bench, and used the machine to provide experimental data for the stability-based stall model which was developed. However, only the first rotor row and the last stator row were fitted to the machine for this experimental work.

Full reblading and recommissioning of the Rofanco test bench was performed by Roos (1990). The performance of the compressor with the new blading was determined, and compared reasonably well with the predictions of Bernad´e (1986). The next developments were in the field of axisymmetric througflow simulations. A new method, the streamline throughflow Method (STFM) was developed to over-come the flaws in the existing throughflow methods, namely the streamline curvature method (SCM) and the matrix throughflow method (MTFM). The first version of this method was developed and described in Roos (1995), and a slightly differently formulated version was compared to the SCM in Gannon and von Backstr¨om (2000). As a result of these efforts, a computer software package implementing the STFM for the axisymmetric simulation of axial flow compressors was written. Thomas (2005) added blade row and boundary layer blockage modelling to this package.

At the same time, another computer software package, the Stellenbosch Univer-sity Compressor Code (SUCC) was being developed. This package contained an implementation of the MTFM, but made allowance for the implementation of alter-native throughflow methods. Both packages were lacking any sort of stall prediction

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capability. The importance of this capability has already been discussed, therefore it was considered necessary to implement a stall prediction model in at least one of the codes. On considering the codes, it was decided to implement the model in the SUCC, as it was written in a modular fashion that simplified this, and was better documented. This project was born of that requirement.

1.2 Problem statement

The Stellenbosch University Compressor Code (SUCC) is a software package in-tended to allow the axisymmetric simulation of axial flow compressor operation for the purposes of performance prediction. A stall prediction model is required for use with SUCC. This model must provide an accurate prediction of the operating conditions which will lead to the onset of compressor stall. This requires that the methods used are able to predict the onset of stall in any of its various forms, such as tip stall or rotating stall, as well as the operational conditions under which a compressor may begin to stall, or experience surge. The resulting stall prediction models should provide a high degree of accuracy in prediction of flow under con-ditions identical to those for which experimentally or practically obtained data is available. It is therefore necessary to determine which of the existing stall predic-tion models is most accurate, and under which condipredic-tions of simulated compressor operation it is most applicable.

1.3 Objectives of this thesis

The main objective of this thesis is:

1 To investigate and evaluate existing stall prediction models for various types of compressor.

The secondary objectives are:

2 To determine what effect the choice of throughflow method has on the accuracy of each stall prediction criterion.

3 To determine the effect of the use of blockage and mixing models, or their omission, on the accuracy of stall prediction for the various criteria.

4 To determine the effect that the layout and coarseness of the computational grid have on the the accuracy of each stall prediction criterion.

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5 To determine which aspects of stall in an axial flow compressor require full three-dimensional treatment. In other words: to determine which aspects of compressor stall cannot be accurately modelled or simulated by use of a two-dimensional blade cascade, and meridional throughflow methods.

1. Evaluation of existing stall prediction models

A number of stall prediction criteria that have been found in literature were imple-mented in the SUCC and used to obtain stall predictions for a variety of compressor test cases.

2. Effect of choice of throughflow method

Stall prediction methods were applied to simulations of test cases using the matrix throughflow method and the streamline curvature method to solve the meridional flow field. The stall predictions for each criterion and each throughflow method were then compared to experimental data from previous researchers to determine the accuracy of the results. It was necessary to implement the streamline curvature method in the SUCC to perform this comparison.

3. Effect of blockage and mixing models

Simulations of the test cases were performed with and without the modelling of effects such as boundary layer blockage and radial mixing. The stall prediction results from these simulations were then compared with experimental results. It was necessary to implement a radial mixing model to perform this comparison.

4. Effect of layout and coarseness of computational grid

An attempt was made to discover the sensitivity of the the stall prediction methods to the number of streamlines in the grid. The effect of the number and placement of quasi-orthogonals was also noted.

5. Limitations of stall prediction in axisymmetric throughflow simulations

Literature describing the various models for the mechanisms involved in the incep-tion and propagaincep-tion of stall cells, and the experimental investigaincep-tion of stall in compressor test benches, was examined. The conclusions and insight drawn from this were useful for the selection or rejection of stall criteria and models for imple-mentation in the SUCC.

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1.4 Concluding remarks

This thesis required the adaptation and understanding of various stall prediction models based upon the work of previous researchers in the field, and the implemen-tation of these models as FORTRAN subroutines in the SUCC. It focuses mainly on flow phenomena involving the rotor blades of axial flow compressors. The inves-tigation was thus largely restricted to blade row and stall related subjects, although it was necessary for the author to become familiar with many other aspects of axial flow compressor analysis and simulation to better understand the workings of the SUCC. Chapter 2 contains descriptions and discusssions of these subjects. Chap-ter 3 describes the implementation of the stall criChap-teria, giving further theoretical background and describing the algorithms used to implement them. Chapter 4 in-troduces the four test cases used to evaluate these stall criteria. The results of the evaluation for each test case, a comparison with experimental data and the simula-tion results of Aungier (2003) and a discussion thereof, are then presented. Chapter 5 provides contains interpretation of the significance of the results obtained and attempts to draw useful conclusions from them. Subjects requiring further research are also described.

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Axial flow compressor theory

This chapter begins by explaining some terminology and theory which is generally applicable to turbomachinery. It then briefly describes some of the axisymmetric throughflow methods used to predict flow in such machinery. Stall is then defined and differentiated from similar phenomena such as surge. Various models for the inception of stall are explained and discussed, and methods for the prediction of the stall limit for a compressor are introduced to the reader. The chapter concludes with a brief overview of the use of CFD analysis in stall prediction and simulation.

2.1 General turbomachinery theory and terminology

The field of axial flow compressor analysis has been extensively explored, and it is not the intention of this author to provide a detailed explanation of the turbomachine theory relating to this type of machine. Readers requiring such a treatment are directed to the many excellent books written on this subject. Those used most extensively by the author were Aungier (2003) and Cumpsty (1989). However, it is desirable to include a very brief examination of some of the more important concepts and definitions from the field in this work, as all subsequent work in this document is somewhat dependent upon them.

Both Aungier (2003) and Cumpsty (1989) use Euler’s turbine equation (2.1) as a starting point for their introduction to basic compressor theory. This equation represents the work done by an idealised turbomachine on the flow passing through it, on a single stream surface.

w = ˙mω (r2Vθ2− r1Vθ1) (2.1)

Equation (2.1) relates the work performed on the flow, w, to the inlet and outlet

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radii of the stream surface (r1 and r2) and the corresponding velocity components in the tangential direction (V_θ₁ and V_θ₂).

Aungier (2003) and Cumpsty (1989) also both present definitions for the flow coefficient φ (2.2) and the stage loading, or stage work coefficient ψ (2.3).

φ = Vx U (2.2) ψ = ∆H U2 = ∆P0 ρω2_D2 (2.3)

The stage loading relates the pressure change across a stage of a compressor to the approximate kinetic energy of the flow relative to the blade, while the flow coefficient is the ratio of the axial component flow rate to the blade tip speed. Another important quantity which is defined in both Cumpsty (1989) and Aungier (2003), is the reaction ratio, R, defined in Equation(2.4), which is a measure of the portion of the static pressure rise for a compressor stage which occurs in the rotor of the stage, i.e.

R = W

2 1 − W22

2U (Wθ1− Wθ2) (2.4)

in which W1 and W2 are the inlet and outlet velocities relative to the rotor blades, and W_θ1 and W_θ2 are the tangential components of the relative velocities. Both Cumpsty (1989) and Aungier (2003) state that it is generally accepted that the optimal value of the reaction ratio is R = 0.5, which indicates an equal pressure rise across stator and rotor of a stage. However Cumpsty (1989) provides examples of compressors for which the reaction ratio does not correspond to this value. This is also the case with two of the test cases used in this project, namely the Rofanco 3-stage low speed compressor and the NACA 5-stage transonic compressor, both of which have a comparatively higher degree of reaction.

A useful quantity which Cumpsty (1989) deals with considerably earlier than Aungier (2003) is rothalpy, I, defined in equation (2.6). The quantity is arrived at by substituting the expression for work performed by the turbomachine according to Euler’s turbomachine equation (2.1) into the energy equation along a streamline crossing a blade row in an adiabatic turbomachine, shown in equation (2.5). This equation is then manipulated and the absolute velocities rewritten in terms of blade speeds and relative flow velocities, so as to reflect the conservation of a quantity that is conserved along a streamline crossing blade rows. For a rotating frame of reference, this quantity is analogous to enthalpy in a static frame of reference, and is thus named rothalpy. It can also be seen that the rothalpy will equal the stagnation

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enthalpy for a static blade row. h1+ V12+ w = h2+ V22 (2.5) (2.6). I = h +W2 2 − U2 2 (2.6)

The diffusion factor, DF , shown in Equation (2.7), is of considerable importance in several stall prediction theories, and is thus of importance in this text. Cumpsty (1989) defines it as follows:

“. . . Essentially this relates empirically the peak velocity on the suction surface of the blade to the velocity at the trailing edge . . . ”

The significance of the diffusion factor is that it gives an indication of the thickness of the boundary layer and likelihood of boundary layer separation occurring on the suction surface of a blade, which would dramatically increase blade profile losses, and could also indicate that the blade (but not necessarily the compressor) was operating in a state of stall.

DF = 1 W₁ µ W1− W2+∆W_2σθ ¶ ≈ Wmax− W2 W₁ (2.7)

Cumpsty (1989) provides a virtually identical expression, but substitutes absolute velocities for the relative velocities used by Aungier (2003). In the approximate equality, given by Aungier (2003), W1and W2are the relative inlet and exit velocities for the blade row, and Wmax is the maximum relative velocity attained in the blade

row ∆W_θ is the change in relative swirl velocity across the blade row. The symbol σ represents the solidity of the blade row, or the ratio of blade chord to circumferential blade pitch for the rotor. Cumpsty (1989) and other sources state that for most axial flow compressors, σ is generally greater than 1 but rarely exceeds a value of 2.

In the early days of axial flow compressor design, it was considered inconvenient to calculate the diffusion factor, as this required the value of Wmax, which was

difficult to obtain. This led to the development of the equivalent diffusion factor,

Deq, which was based upon the geometric properties of the blade row. The equivalent

diffusion factor is defined in equation (2.8):

Deq= Wmax

W2

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Cumpsty (1989) provides equation (2.9) for the calcualtion of Deq: Deq= cos κ_{cos κ}2 1 " 1.12 + a (∆i)1.43+ 0.61cos2κ1 σ (tan κ1− tan κ2) # (2.9) The formulation presented by Aungier (2003) is virtually identical, except that it is written in terms of flow angles β1 and β2 instead of blade metal angles κ1 and κ2. The diffusion factor will be discussed further when the associated stall prediction method is explained in the next chapter.

2.2 Numerical simulation of compressor operation

2.2.1 Introduction

As was mentioned in chapter 1, the increasing power of computer hardware, coupled with advances in the field of CFD make it theoretically possible to perform a full three-dimensional simulation of the operation of an axial flow compressor. However, such simulations are still rather costly in terms of the time required to prepare them, the simulation execution time, and the actual financial cost of the software. Because of their speed and relative simplicity, the meridional analysis methods are therefore still widely used for purposes where they offer sufficiently accurate results.

Three meridional analysis methods will be discussed. All are inviscid flow ap-proximations which assume axisymmetric flow in the compressors which they are used to simulate. They generally make use of boundary layer assumptions and smoothing mixing models to approximate endwall blockage and radial mixing, which are not accounted for by inviscid flow methods.

Another important simplifying assumption which can be made is that the ef-fects of the various blade rows can be approximated by one or more actuator disks per blade row. Actuator disks can be regarded as infinitely thin rotors having no frictional losses across them, with an infinite number of blades, which impart or remove energy and angular momentum to or from the flow. An improvement of the accuracy of the simulation can be obtained by calculating the exit angle of the flow from the blade row based upon blade row geometry and the deviation angle for the blade row obtained from empirical relations, usually obtained from experimental blade cascades. The relative outlet velocity vector for the blade row can then be determined. Finally, the blade row speed is subtracted so as to yield the absolute flow velocity vector. Because the phenomenon of stall is strongly linked to blade row geometry by almost all researchers, the latter model was the one used in all simulations in this project.

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Of the three throughflow methods, the streamline curvature method (SCM) will be discussed in the greatest detail, as it was necessary to implement it in order to complete this project. The matrix throughflow Method (MTFM) will also be discussed in some detail, as it was the throughflow method originally implemented in the SUCC. The streamline throughflow method (STFM) will be briefly discussed as it was locally developed, and will almost certainly be implemented in the SUCC by future researchers.

2.2.2 Streamline curvature method

The first method to be discussed streamline curvature method (SCM). This method appears to be by far the most popular, judging by comments made in Cumpsty (1989), the fact that no other throughflow method is described by Aungier (2003), and that it is used as a standard against which to measure other throughflow meth-ods, as in Gannon and von Backstr¨om (2000). In this method, the meridional flow field is divided up into a computational grid by a number of quasi-normal lines (i.e. lines approximately normal to the streamlines for the flow field within the compres-sor), and streamlines, the positions of which are initially estimated based on equal fractions of the annulus radius or some similar crude method, and change as the simulation progresses. The quasi-normals, or quasi-orthogonals are usually oriented approximately radially. They are often positioned on the leading and trailing edges of the blade rows, and their position remains fixed throughout the simulation. The intersections between the quasi-normals and the streamlines form the computational grid for the streamline curvature method. Starting with the first quasi-normal at the compressor inlet, and moving towards the exhaust, the meridional flow field on each quasi-normal is solved at each grid point by iteratively performing a mass-balance and a momentum-balance on the meridional flow across the quasi-normal. The momentum-balance is performed by solving a discretized form of equation (2.10), which is the axisymmetric, time-steady, normal momentum equation for flow in the compressor. This requires that the curvature for each streamline be calculated at each grid point on the quasi-normal, giving the method its name. The swirl compo-nent of the velocity is calculated using either actuator disk models, or from empirical blade row models based on experimental cascade data. The other properties of the flow, for instance the temperature and entropy of the gas, are then calculated at each grid point on the quasi-normal. Finally, once the meridional flow field has been solved on all quasi-normals, new values for the positions of the streamlines on the quasi-normals are calculated. The change in streamline position must be under-relaxed in order to ensure stability of the method. The entire process is repeated

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until the position of the streamlines change less than a certain desired tolerance. Equation(2.10), the momentum equation for the normal direction as given by Cumpsty (1989), is thus the basis of the streamline curvature method:

1 2 ∂ ∂yV 2 m= ∂H ∂y − T ∂s ∂y + Vm ∂Vm ∂m sin (φ + γ) + V2 m r_m cos (φ + γ) − 1 2r2 ∂ ∂y ³ r2V_θ2´+ Vm r ∂ ∂m(rVθ) tan ² (2.10)

In Equation (2.10) φ is the angle subtended by a vector in the axial direction, x, and the vector m, which is tangential to the streamline. The angle γ is the angle subtended by the radial direction r vector and the quasi-normal q. The angle ² is the angle between the meridional plane and the curved surface upon which the streamlines are to be found. The symbol H is the stagnation enthalpy of the flow. This information is derived from a description of the method in Cumpsty (1989). A similar explanation is provided by Aungier (2003), who bases his version of the SCM on that of Novak (1967). Cumpsty (1989) and Roos (1995) explain that an advantage of the SCM over stream function based throughflow methods such as the MTFM and the STFM is that it can handle areas of supersonic meridional flow, which would imply two possible solutions for the flow field for a stream function based method. The main disadvantage is its inherent instability, which requires the use of numerical damping. A more detailed explanation of the SCM, as described in Aungier (2003) and implemented during this project, is included in appendix B.

2.2.3 Matrix throughflow method

The second method to be discussed is derived from a similar theoretical basis to the SCM. It is called the matrix throughflow method (MTFM). The method makes use of the concept of the stream function, represented by the symbol ψ. By definition, this function has a constant value along a streamline or stream surface. The velocity component along a direction vector at a given point in a flow field can also be determined if the gradient of the stream function with respect to the direction vector is known. Thus, if the value of the stream function is known at various points of a flow field (for instance, in a relatively evenly distributed grid across the entire flow field), the flow velocity and direction can be determined at any point in the flow field by means of interpolation and numerical differentiation. Thus the meridional flow field is divided up by streamlines and quasi-normals to form a computational grid in the same way as is done for the streamline curvature method. An important difference between the two methods is that for the matrix throughflow method, the positions of the streamlines used to define the computational grid are not adjusted

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from their initial estimated positions during computation. For this reason they are called quasi-streamlines. The position of the true streamlines may be calculated by interpolating in terms of the stream function between the grid points on the computational grid.

The value of the stream function at various points in the flow field can be solved by discretizing equation (2.11) and solving the resulting system of equations in ψ at each point on the computational grid by means of a matrix inversion. Equation (2.11), also given by Cumpsty (1989), is derived from the same momentum and en-ergy equations which give rise to equation(2.10) of the streamline curvature method, but it is reformulated in terms of the stream function.

∂2ψ ∂p2 + ∂2ψ ∂y2 = ρ rB Vp µ ∂H ∂y − T ∂s ∂y − Vθ r ∂ ∂y(rVθ) + aθtan ² ¶ + ∂ψ ∂p ∂ ∂pln (ρrB) + ∂ψ ∂y ∂ ∂yln (ρrB) (2.11)

Thus the values of the stream function at each point on the computational grid are obtained at once. However, examination of equation (2.11) will reveal that the solu-tion requires the values of the velocity components and thermodynamic properties on the computational grid. It is therefore necessary to begin with estimated values and improve these values by successive applications of the method. In the SUCC, the process is stabilised somewhat by beginning the simulation at lower rotational speed and a correspondingly low massflow rate, or a solution for a previous sim-ulation with similar operating conditions, and slowly increasing it to the desired simulation conditions.

This information is derived from the description of the matrix throughflow Method given in Cumpsty (1989). A far more complete explanation of the ma-trix througflow Method and its derivation is to be found in Thiart (2005), which is included as appendix A.

2.2.4 Streamline throughflow method

The third method of determining the shape of the streamlines within the compres-sor is known as the streamline throughflow method. This axisymmetric method is the most recent of the three methods discussed here, as it was developed at the University of Stellenbosch in the 1990’s. The development of this method, and the advantages that it offers over the SCM and MTFM are described in detail in Roos (1995) and Gannon and von Backstr¨om (2000). Here follows a brief description of this method, as gleaned from these two sources.

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Boadway’s transformation is then applied to the fundamental equations to yield the partial differential equation of the form shown in (2.12):

Ã 1 + µ ∂r ∂z ¶₂! 1 ρr ∂ ∂ψ µ ρr∂r ∂ψ ¶ + µ ∂r ∂ψ ¶₂ ρr ∂ ∂z µ 1 ρr ∂r ∂z ¶ = 2∂r ∂z ∂r ∂ψ ∂2_r ∂z∂ψ − (ρr) 2µ∂r ∂ψ ¶₃· ∂h0 ∂ψ − T ∂s ∂ψ − Cθ r ∂ (rCθ) ∂ψ ¸ (2.12) This equation is then solved numerically for a radius, r, in terms of the known quantities stream function ψ and axial position z by means of a matrix inversion. In other words, instead of calculating the value of the stream function at various points on a computational grid, then interpolating between points to find the position of the streamlines, as in the MTFM, the STFM assumes certain values for the stream function, then calculates the radial positions of streamlines at each axial position.

Gannon and von Backstr¨om (2000) provides a good summary of the advan-tages of the streamline throughflow method over the older streamline curvature method. Perhaps the most important advantage is that it inherently satisfies the mass-conservation criterion, which eliminates the need for an iterative process to ensure this. As a consequence, a computational solution utilising this method con-verges to a solution in an order of magnitude less time than an equivalent SCM computational solution, Other advantages are that this method is somewhat more numerically stable and more tolerant of distorted computational grids. It produces results of a level of accuracy very similar to that achieved by the streamline curvature method and the matrix throughflow method.

2.2.5 Blade cascade modelling

An approximation often used in modelling of axial flow turbomachines is to re-gard the flow along a stream surface through the blades in a blade row as a two-dimensional flow. Such an arrangement is termed a blade cascade.

Figure 2.1, copied from a very similar figure in Aungier (2003), shows the most important geometric properties for blade calculations. The mean camber line is a line connecting the origins of the set of all inscribed circles between the blade row surfaces. The blade chord line is a line joining the points where the leading and trailing edges intersect the mean camber line. The angles κ1 and κ2 are termed the inlet and outlet blade metal angles, and are the mean leading and trailing edge angles of the blade profile relative to the axial direction. Under design conditions, flow relative to the blade row would enter and leave the cascade at these respective angles, if there were no flow deviation. The actual relative velocities, W1 and W2,

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at which the flow enters and leaves the cascade make relative angles of β1 and β2 with the axial direction. The difference between the blade metal inlet and the flow inlet angles, κ1 and β1, is termed the angle of incidence, i. This is usually used as a parameter in blade cascade models, although some require the angle of attack,

α, which is defined as the angle between the flow and the blade chord line. The

difference between the outlet metal and flow angles is termed the deviation angle,

δ, and must be accounted for by a blade cascade correlation, usually empirically

based, to be described shortly. The difference between κ1 and κ2 is termed the

camber angle, θ, and the angle between the line joining the leading and trailing

edges and the axial direction is termed the stagger angle, γ.

Aungier (2003) provides a method for calculating blade cascade losses, loadings and deviation angles. The first step is to calculate the design angles of attack (α∗₎

and incidence (i∗_{). using an iterative scheme, since both are dependent on one}

another. Aungier (2003) provides correlations that are obtained by fitting curves to the data obtained by Johnsen and Bullock (1954). Aungier then calculates the design deviation angle, δ∗_{, using correlations developed by Lieblein, described in}

Johnsen and Bullock (1954). Following this, Aungier (2003) presents a method to calculate the pressure loss coefficient for flow at the design incidence angle, ω, once again developed by Lieblein. This requires the calculation of the equivalent diffusion factor, Deq. The loss coefficient thus calculated only accounts for profile

losses, so additional loss models must be used to account for other losses due to blade tip leakage, Mach-number related effects, endwall boundary layer effects, and leakage through seals. The originator of the method which Aungier (2003) advocates

γ α W1 β2 β₁ κ i s W1 c W2 δ κ2 1

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was Howell (1945). The loss coefficient is then adjusted to represent off design conditions. Once all the loss-coefficients have been calculated, they can be added together. Aungier (2003) recommends that the limit ω ≤ 0.5 be applied to all loss coefficients. The loss coefficient is used to calculate the pressure loss over the blade row on each stream-surface examined by use of equation(2.13):

(∆Pt,rel) = ω (Pt− P )in (2.13)

Finally, for a compressor operating in transonic conditions, shock losses must be accounted for.

The details of the method presented here in overview are to be found in Aungier (2003), and a similar but alternative method was implemented by Thomas (2005). Some details of the implementation of these models in SUCC are given by Thiart (2004).

2.2.6 Radial mixing modelling

An assumption inherent in all the meridional throughflow methods discussed is that no flow occurs along quasi-normals, or, in other words, no mixing of flow occurs between stream surfaces. Cumpsty (1989) and Aungier (2003) state that this as-sumption will lead to an unrealistic loss or entropy distribution along quasi-normals in blade rows, as mixing does occur between stream sheets in a real compressor. As quasi-normals are usually approximately radially aligned for an axial flow compres-sor, the mixing between stream sheets is often referred to as radial mixing. Both Aungier (2003) and Cumpsty (1989) agree that the omission of some correction to allow for radial mixing in throughflow methods can cause numerical instability in the throughflow method to the extent that the method will not converge to a solu-tion. Aungier (2003) adds that the errors resulting from the omission of a mixing model will be most severe when the simulation is being performed for conditions close to that of the surge limit of the real machine. Cumpsty (1989) provides an explanation for radial mixing based on that of Gallimore and Cumpsty (1986) which accounts for this. In this model, mixing is equated to turbulence, which Cumpsty (1989) defines for this purpose as any non-deterministic, unsteady flow occurring within the machine. Now the onset of surge is determined by the stability of the compressor system, according to Pampreen (1993) and Cumpsty (1989), as will be discussed in the next section. As the real compressor approaches the point of surge, the stability of the flow will break down, and more and more unsteadiness will be observable in the flow, thus a greater degree of mixing will occur.

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Cumpsty (1989) also provides an alternative model for radial mixing based upon the work of Adkins and Smith (1982). In this model, the mixing is assumed to be caused by secondary flows in a radial direction, due to effects such as non-uniform blade circulation, radial flow in the boundary layers, tip leakage, and leakage from the suction to the pressure surface of a blade near the endwalls. In other words, the mixing is assumed to be caused by secondary flow patterns of deterministic (and thus predictable) origin. This model for the smoothing of of a general thermodynamic property P . The mixing is simulated by solution of equation(2.14):

∂P ∂z = β

∂2_P

∂r2 (2.14)

In equation(2.14, the value of the mixing parameter β is given by:

β = x a Z _a 0 V2 r V2 z da

Aungier (2003) suggests that radial mixing be approximated by applying smooth-ing to the total pressure losses calculated for each blade row, (∆P0

t)i, according to

the weighted averaging scheme described in equations (2.15) to (2.17). The principle behind this scheme is that the value of a numerical approximation of the hub-to-tip integral of the property will be identical for the original and smoothed values. The symbol i denotes the number of the streamline on which the smoothing is applied, with 1 at the hub, and N at the tip.

¡

∆P_t0¢_i,smooth =h¡∆P_t0¢_i−1+ 2¡∆P_t0¢_i+¡∆P_t0¢_i+1i/4;1 < i < N (2.15)

¡

∆P_t0¢_1,smooth = 2¡∆P_t0¢_2,smooth−¡∆P_t0¢_3,smooth (2.16) ¡

∆P_t0¢_N,smooth = 2¡∆P_t0¢_{N −1,smooth}−¡∆P_t0¢_{N −2,smooth} (2.17) This method of modelling radial mixing is rather crude, as it is not based upon a specific explanation of the mechanism leading to radial mixing. It merely seeks to mimic the effects of radial mixing to some degree. Another criticism is that the degree of smoothing is dependent to some extent on the number of iterations, as the smoothing is applied for each iteration of the throughflow method. Of course, the properties which are smoothed are recalculated during each iteration, but these new values will be based partially on the old values of the various properties, which

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have already been smoothed. The chief advantage of this method is that it is very easy to implement.

2.3 The nature of stall

2.3.1 Stall, surge and choke

The terms “stall” and “surge” are often confused or used interchangeably when used in the study of axial flow compressors, although they are rather different in nature. Generally, one attempts to predict the conditions under which stall will occur, as it will usually precede surge, and is slightly less dangerous to compressor hardware. It is possible for a compressor to operate within its design envelope with minor stall, or small stall cells occurring in some blade rows. Once a compressor enters a state of surge, however, all normal, steady operation ceases. Here follows a brief explanation of these two phenomena, deriving from the explanations provided by Cumpsty (1989) and Pampreen (1993).

In axial flow compressors, stall is a condition whereby flow through the machine is partially blocked by one or more pockets of the working fluid not moving in the intended direction through the blade passages in one or more rotor or stator rows. Such a pocket of fluid is referred to as a stall cell. boundary layer separation on the hub or shroud can influence the formation of stall cells and the onset of stall. A compressor will experience a large drop in discharge pressure when entering stall, but may continue operating in a relatively stable fashion in this new condition. However, the performance is thus obviously degraded, and the stall cell rotates relative to the blade row it affects, at a significant fraction of the rotational speed of the compressor. This, coupled with the unsteady nature of flow within the stall cell can cause the individual blades in the blade row to vibrate, which can lead to fatigue-related failures, which can have catastrophic results. It is thus highly undesirable for an axial flow compressor to operate stalled. There is also the danger that if an axial flow compressor is sufficiently badly stalled, and a sufficiently large difference in pressure exists between the discharge end and the inlet, surge may occur.

Compressor surge is defined as a total breakdown of stable compressor operation. Whereas most of the fluid in a stalled compressor will still be travelling from the inlet to the exit, this is not the case for a machine suffering from surge. Under this condition, the flow and pressure gradient within the compressor may change rapidly and unexpectedly, and flow conditions within the machine bear no resemblance to those for which it was designed. The condition is thus highly undesirable, as the

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various components are exposed to conditions for which they were not designed. Surge is a system phenomenon; that is to say that it is dependent not only upon the properties of the compressor, but of the entire flow system to which the com-pressor is coupled. The frequency of the surge fluctuations is dependent on both the compressor and the volume of the ducts or chambers from which the compressor receives and to which it exhausts working fluid.

It is simpler to predict stall than surge, as it is necessary to know something of the system of which the compressor forms a part in order to arrive at an accurate surge prediction, as surge is a system-dependent phenomenon, while it is only necessary to know the operating conditions and geometry of the compressor being considered in order to predict stall.

At the opposite extreme to surge and stall, a compressor may operate in a choked condition. Choke in a compressor is very similar to choke in a duct; that is to say that it represents the maximum flow rate attainable by the working fluid through the blade row passages, and generally implies that the average throughflow velocity is equal to the local speed of sound in the fluid based upon the relative stagnation properties of the fluid upstream of the choke location.

2.3.2 Cumpsty’s discussion of compressor stall

Cumpsty (1989) and Pampreen (1993) differentiate between progressive and abrupt stall, and full-span or part-span stall. Progressive stall is normally part span, and occurs near the tips of the stalled blade row, while abrupt stall is usually full-span. Abrupt, full-span stall is an extremely serious condition, which can do significant damage to a compressor because although the pressure rise and gas flow rate are generally low, the work input to the compressor is high under such a condition, and this work is transformed into heat energy, causing the compressor to become dangerously hot, and possibly suffer thermally-related damage.

Cumpsty (1989) provides a simplified explanation of how a stall cell forms and grows. Excessive flow separation occurs in one or more blade passages in the rotor, leading to the formation of a small stall cell in that passage. The separation may be caused by a tiny imperfection in the surface or alignment of one of the blades. What-ever agency causes the separation to occur, it increases in sWhat-everity until it effectively blocks the flow through this passage, which affects the flow through neighbouring passages. The blades ahead of the stalled blade in the direction of rotation of the rotor are subjected to flow at a lower angle of incidence than they were designed for, which takes the stalled blade passage out of stall, while the angle of incidence of those following increases, causing them to stall as well. This causes the stall cell to

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grow somewhat, and rotate around the axis of the machine, in the same direction as the rotor, only more slowly, once it has grown as much as it can. This means that the stall cell rotates backwards relative to the rotor. This is illustrated in figure 2.2 Cumpsty (1989) also mentions the phenomenon of hysteresis, which a compressor will exhibit while stalling and recovering from stall. A compressor which has been throttled to the point of stall inception and beyond will not recover from stall until the throttle has been opened significantly further than the position at which it was set when stall initiated. In other words, the compressor leaves the stalled condition at a considerably higher massflow rate than that at which it enters the stalled condition. This can have serious implications on stall-recovery in compressors, particularly in aero-engines, where the hysteresis effect has been known to make it necessary for an engine to be shut down and restarted in order to recover from stall.

Cumpsty (1989) explains that a compressor with low aspect ratio, long chord blades stalls at a much lower flow coefficient than one with high aspect ratio blades, as the former are less badly affected by the pressure and velocity gradients near the endwalls of the compressor. The former are also more resistant to vibration than the latter.

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2.3.3 Day’s findings on stall inception mechanisms

I. J. Day is a researcher who has made an important contribution to the current understanding of rotating stall phenomena in axial flow compressors. In Day (1993) he discusses two of the most common flow models used to explain the inception of rotating stall in an axial flow compressor. The first he discussed is essentially the same as the explanation given by Cumpsty (1989) in the previous section, involving the separation of flow from a single blade, the resulting blockage of the blade passage, and the resulting alteration of the angle of incidence of the blades to either side of the blocked passage, causing the stall cell to move around the blade row. This explanation dates back to the 1950’s. He then draws attention to the fact that a stall cell generated by this mechanism will only affect a few blades at a time, and will thus be small. Such a stall cell will also only affect one blade row. For this reason he refers to this type of stall inception mechanism as a short length scale

disturbance.

The second flow model which Day (1993) discusses is of more recent vintage, and was first developed by Moore and Greitzer (1986a) and applied in Moore and Greitzer (1986b). This model supposes a sinusoidal velocity variation about the annulus of the compressor, affecting flow throughout its entire length, which rotates and has a wavelength equal to the circumferential distance around the annulus. The speed at which the velocity-variation rotates is smaller than that of the blade rotation, but it occurs in the same direction. This velocity variation is initially very small, but grows rapidly in amplitude as the stability limit of the compressor is approached, and would, according to the theory, thus bring about stalling in the machine in an apparently instantaneous fashion. Some practical measurements, particularly by McDougall et al. (1990), indicate that these modal perturbations, as Day calls them, do indeed exist and play a role in stall inception.

By experimental techniques described in the next section, Day demonstrated that modal velocity disturbances are not a mechanism leading to the inception of stall, but rather an indication that the compressor is nearing its region of unstable operation. An alternative indication is the formation of a small, but finite sized stall cell. He explains that while a modal disturbance is a relatively reversible type of disturbance, which need not seriously affect the compressor performance, the formation of a small finite stall cell is highly irreversible, and will precipitate stall in a comparatively short time. He adds that coupling can only occur if the modal disturbances are well established by the time of the formation of the finite sized stall cell.

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circumference of the annulus, or occur around a larger portion of the circumference, giving rise to the perception that there are ‘small’ and ‘large’ stall cells, although in fact stall cells could occur in any intermediate size. He states that the smaller, more restricted stall cells move faster around the annulus than the larger, less restricted ones, and that their onset is more sudden and violent. The formation of a finite sized stall cell destroys any modal disturbance that is not already well established, and interacts with those that are. Finally, Day (1993) states that the smaller, short length scale stall cells appear to be the most common cause of stall inception in the compressor.

Day continued his research into short length scale and modal disturbance phe-nomena, and in Day and Camp (1997) presented further findings obtained in con-junction with Camp. His findings were that a modal disturbance of sufficiently large amplitude could result in a breakdown of flow through the compressor in the part of the compressor annulus affected by the velocity trough. This could lead to the gradual, slow formation of a large, slowly rotating stall cell, although usually a spike-disturbance, or short length scale stall cell, developed first.

2.4 Experimental investigation of stall

Many important experimental investigations into the onset of stall, and its causes, were performed by I. J. Day, and was described in Day (1993). A brief account of his experimental approach and subsequent findings are included below, as they provided a considerable insight into the nature of stall, particularly the phenomenon of rotating stall.

Day conducted a practical experimental investigation into the role played by modal disturbances by measuring flow velocities as a function of time at various angular positions at a constant radius and axial position ahead of the first rotor row in two different axial compressors entering stall; one a single-stage machine, and the other a multi-stage machine. From the time-dependent data gathered, the inception, growth and speed of rotation of an instability about the axis of the machine in an angular direction could be determined.

Day then conducted a series of tests in which he artificially created modal dis-turbances by injecting high speed air jets into the compressor inlet at various radial positions while the machine was running at the lowest throughflow velocity possible without initiating premature stalling. The frequency of the induced disturbance was varied by control of these jets, and the amplitude of the variation of the velocity at a later stage in the compressor was recorded and plotted against the frequency of the induced disturbance. The range of frequencies in which a maximum variation of

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flow velocity was recorded corresponded remarkably well with the rotational speed of the disturbance measured in the previously described test.

Day then proceeded to investigate short length scale stall cells and their forma-tion. His eventual findings were that short length scale stall cells move considerably faster around the compressor annulus than larger ones, and generally prevent stall cells due to modal disturbances from forming, as they destroy the axisymmetric na-ture of the modal oscillations. He also concluded that short length scale disturbances were a more common cause of stall inception than modal disturbances.

A number of researchers have since attempted to find indicators for the onset of stall of various types in axial flow compressors. Many of the stall prediction criteria which have now achieved general acceptance had their basis in correlations drawn from experimental data rather than theoretical flow analysis.

Many recent researchers have resorted to sophisticated measuring devices, and the development of new methods of practically obtaining stall measurement is a rich field of development and research.

Both Inoue et al. (2001) and Bernd H¨oss and Fottner (2000) adopted an ap-proach involving the use of Fourier-transforms and Wavelet-transforms on pressure fluctuations in axial flow compressors. The latter was interesting in that the exper-imental data was obtained from an entire working turbofan engine, rather than a compressor operating as an isolated test bench.

Ottavy et al. (2001) presented a two-part paper that described the analysis of flow fields between the blade rows of a transonic axial flow compressor. The first part describes the experimental method employed to obtain a velocity distribution, using the technique of Laser 2-Focus anemometry. This measurement technique is described at some length, followed by a description of the moving shock observed within the compressor under investigation. The second part of the paper consisted of an explanation of an unsteady flow analysis performed on the compressor which was tested. The analysis compared data obtained experimentally for inlet guide vane wakes and rotor leading edge shocks with those obtained from an analytical model, with good agreement reported between the two.

Locally, Lewis (1989) made use of the Rofanco 3-stage low pressure compressor test bench at Stellenbosch University to validate a small perturbation stability stall inception model. The stall model in which the experimental data was used was based upon a stability analysis of the flow field between the trailing edge of a rotor blade row and the leading edge of the stator in the same stage. As a result, only the first stage rotor blade row and the last stage stator blade row were present in the machine during experimental testing, so as to provide a large blade free region. A 3-hole cobra probe was used to measure flow properties within the compressor, as it

A comparison between stall prediction models for axial flow compressors

Compressors

Andrew Gill

Declaration

Abstract

Opsomming

Acknowledgements

Contents

List of Figures

List of Tables

Nomenclature

Introduction

1.1

Background

1.2

Problem statement

1.3

Objectives of this thesis

1.4

Concluding remarks

Axial flow compressor theory

2.1

General turbomachinery theory and terminology

2.2

Numerical simulation of compressor operation

2.3

The nature of stall

2.4

Experimental investigation of stall